Spin-polarized scanning tunneling microscopy of half-metallic ferromagnets:
Non-quasiparticle contributions
V. Yu.
Irkhin
arXiv:cond-mat/0512245v1 [cond-mat.str-el] 12 Dec 2005
Institute of Metal Physics, 620219 Ekaterinburg, Russia
M. I. Katsnelson
Institute of Molecules and Materials, University of Nijmegen, 6525 ED Nijmegen, The Netherlands
The role of the many-body (spin-polaronic) effects in the scanning tunneling spectroscopy of halfmetallic ferromagnets (HMF) is considered. It is shown that the non-quasiparticle (NQP) states
exist in the majority or minority spin gap in the presence of arbitrary external potential and,
in particular, at the surfaces and interfaces. Energy dependence of the NQP density of states is
obtained in various models of HMF, an important role of the hybridization nature of the energy gap
being demonstrated. The corresponding temperature dependence of spin polarization is calculated.
It is shown that the NQP states result in a sharp bias dependence of the tunneling conductance near
zero bias. Asymmetry of the NQP states with respect to the Fermi energy provides an opportunity
to separate phonon and magnon peaks in the inelastic spectroscopy by STM.
I.
INTRODUCTION
The history of the investigations of half-metallic ferromagnets (HMF) starts from the electronic structure calculation
for NiMnSb [1]; later a number of other examples were discovered, e.g., CrO2 , Fe3 O4 , a number of the Heusler alloys
Co2 MnZ and RMnSb (for a review, see Refs. 2, 3). These substances have metallic electronic structure for one
spin projection (majority- or minority-spin states), but for the opposite spin direction the Fermi level lies in the
energy gap. Owing to this fact HMF attract now a growing attention of researchers in connection with “spintronics”,
or spin-dependent electronics [4]. The spin-up and spin-down contributions to electronic transport properties have
different orders of magnitude, which can result in a huge magnetoresistance for heterostructures containing HMF [2].
Some evidences of the HMF behavior in colossal magnetoresistance (CMR) materials like La1−x Srx MnO3 found by
using tunneling spectroscopy [5, 6] and photoemission technique [7] have increased considerably the interest in the
half-metallic ferromagnetism; however, the situation in the CMR systems is controversial, as demonstrate Andreev
reflection experiments [8].
Peculiar band structure of HMF results in an important role of incoherent (non-quasiparticle, NQP) states which
occur because of correlation effects [2]. The appearance of NQP states in the energy gap near the Fermi level
[9, 10, 11, 12] is one of the most interesting correlation effects typical for HMF. The origin of these states is connected
with “spin-polaron” processes: the spin-down low-energy electron excitations, which are forbidden for HMF in the
one-particle picture, turn out to be possible as superpositions of spin-up electron excitations and virtual magnons. The
density of these states vanishes at the Fermi level EF for temperature T = 0, but increases drastically at the energy
scale of the order of a characteristic magnon frequency ω. The existence of NQP states is relevant for spin-polarized
electron spectroscopy [12, 13], NMR [14], core-level spectra of the HMF [15], and subgap transport in ferromagnetsuperconductor junctions (Andreev reflection) [16]. Several experiments could be performed in order to clarify the
impact of the NQP states on spintronics. In particular, I − V characteristics of half-metallic tunnel junctions for the
case of antiparallel spins are completely determined by NQP states [17, 18]. Recently the density of NQP states has
been calculated from first principles for a prototype HMF, NiMnSb [19], and for CrAs [20].
On the other hand, HMF are very interesting conceptually as a class of materials which may be convenient to treat
many-body solid state physics that is essentially beyond band theory. It is accepted that usually many-body effects
lead only to renormalization of the quasiparticle parameters in the sense of Landau’s Fermi liquid (FL) theory, the
electronic liquid being qualitatively similar to the electron gas (see, e.g., Refs.21, 22). On the other hand, NQP states
in HMF are not described by the FL theory. As an example of highly unusual properties of the NQP states, we note
that they can contribute to the T -linear term in the electron heat capacity [12, 23], despite their density at EF is zero
for T = 0.
Spin-polarized scanning tunneling microscopy (STM) [24] is a very efficient new method which enables one to probe
directly the spectral density with spin resolution in magnetic systems. The spin-polarized STM should be able to
probe the NQP states via their contribution to the differential tunneling conductivity dI/dV . At zero temperature,
NQP states arise only above EF for the case of minority-spin gap and only below EF for the majority-gap HMF
[2]. Unlike the photoemission spectroscopy which probes only occupied electron states, STM detects the states both
above and below EF , depending on the sign of bias.
Theoretical investigation of NQP contributions to STM spectra is the aim of the present paper. The paper is
2
organized as follows. In Sect.2 we discuss a general expression for the tunneling current I as applied to HMF. In
Sect.3 the effect of surface potential and other spatial inhomogeneities on the NQP spectral density is considered. In
Sects.4 and 5 we calculate the energy and temperature dependences of dI/dV and treat the problem of tunnelingcurrent spin-polarization at finite temperatures. In Sect.6 the bias dependences of the tunneling conductance are
discussed.
II.
CALCULATION OF THE TUNNELING CURRENT
A general expression for the tunneling current in the lowest order in the tunneling matrix elements has the form
[25, 26]
Z
πe X
σ 2
I=
|M |
dEAσn (E) Aσν (E − eV ) [f (E − eV ) − f (E)]
(1)
~ nνσ nν
where e is electron charge, σ is the spin projection, V is the bias, f (ε) is the Fermi distribution function, Greek
(Latin) indices label electron eigenstates for the sample (tip) ψnσ , ψνσ
Z
~2
∗
∗
σ
dA (ψnσ
∇ψνσ − ψnσ
∇ψνσ )
(2)
Mnν
=
2m
is the current matrix element, m is the free electron mass (the surface integral in Eq.(2) is taken over arbitary area
between the tip and the sample), and
1
Aσν (E) = − ImGσν (E)
π
(3)
is the corresponding spectral density, Gσν (E) = Gσνν (E) is the sample Green’s function,
Gσνλ (E) = hhcνσ |c†λσ iiE ,
c†νσ being the creation operators for conduction electrons. It is worthwhile to emphasize that the expression (1) takes
into account correlation effects in a formally exact way, assuming that the tunneling probability is small. In fact,
the latter condition should be satisfied for proper STM measurements, otherwise they cannot be considered as a true
probe. In the WKB approximation Eq.(1) takes the form [27]
!
r
2 X
Z
2mΦσ
πe ~2
σ
Nt (EF ) dE [f (E − eV ) − f (E)] gsσ (E)
(4)
I (z, V ) ≃
exp −2z
2
~ 2m
~
σ
where Φσ is the average of sample and tip work functions (which is assumed to be large in comparison with eV for
simplicity), z is the distance between the surface and the tip. Here Nt (E) is the density of states (DOS) of the tip
material, which is supposed to be smooth and thus is replaced by its value at the Fermi energy, and
X
gsσ (E) =
(5)
Aσν (E) hνσ| δ kk |νσi
ν
is the density of states of the sample with zero in-plane component of the wave-vector: kk = 0 so that the summation
is performed only over two points of the Fermi surface. This condition means that the tunneling probability has
a sharp (at not too small z) maximum for the states with velocity direction normal to the surface. For a generic
multi-sheet Fermi surfaces the condition of the dominant tunneling is, generally speaking, more complicated, but this
modifies only some weakly bias-dependent factors.
III.
NON-QUASIPARTICLE STATES IN INHOMOGENEOUS MATERIALS
Since STM probes only surface one has to discuss first modification of NQP states in comparison with the case of
ideal bulk crystal. The existence of NQP states at the surface of HMF has been demonstrated for a narrow-band
Hubbard model [28]. Here we present a general derivation valid in the case of arbitrary inhomogeneity.
3
To describe the effects of electron-magnon interaction for the inhomogeneous case we use the formalism of the
exact eigenfunctions developed earlier for the impurity-state problem in a ferromagnetic semiconductor [29]. The
corresponding Hamiltonian of the s − d exchange model reads
!
Z
X
X
σ
†
†
H =
dr
Ψσ (r)H0 Ψσ (r) − I
δS(r)Ψσ (r)σ σσ′ Ψσ′ (r) + Hd
σ
σσ′
2
H0σ = −
~
∇2 + Uσ (r)
2m
(6)
where Uσ (r) is the potential energy (with account of the electron-electron interaction in the mean field approximation)
which is supposed to be spin dependent, Ψσ (r) is the field operator for the spin projection σ, S(r) is the spin density of
the localized-moment system, δS(r) = S(r)− hS(r)i is its fluctuating part, the effect of the average spin polarization
hS(r)i being included into Uσ (r). We use an approximation of contact electron-magnon interaction described by the
s − d exchange parameter I,
X
Hd = −
Jq Sq S−q
(7)
q
is the Heisenberg Hamiltonian of localized spin (for simplicity we neglect the inhomogeneity effects for the magnon
subsystem).
Passing to the representation of the exact eigenfunctions of the Hamiltonian H0σ ,
H0σ ψvσ = ενσ ψνσ ,
X
Ψσ (r) =
ψνσ (r) cνσ ,
(8)
ν
one can rewrite the Hamiltonian (6) in the following form:
X
X
H=
ενσ c†νσ cνσ − I
(να, µβ|q) δSq c†να σ αβ cµβ + Hd
νσ
(9)
µναβq
where
(νσ, µσ ′ |q) = hµσ ′ | eiqr |νσi .
We take into account again the electron-spectrum spin splitting in the mean-field approximation by keeping the
dependence of the eigenfunctions on the spin projection.
We restrict ourselves to the spin-wave region where we can use for the spin operators the magnon (e.g., DysonMaleev) representation. Then we have for the one-electron Green’s function
−1
Gσν (E) = [E − ενσ − Σσν (E)]
,
(10)
with the self-energy Σσν (E) describing correlation effects.
We start with the perturbation expansion in the electron-magnon interaction. To second order in I one has
Σσν (E) = 2I 2 SQσν (E)
(11)
with
Q↑ν (E) =
X
µq
| (ν ↑, µ ↓ |q) |2
X
Nq + n↓µ
1 + Nq − n↑µ
, Q↓ν (E) =
| (ν ↓, µ ↑ |q) |2
E − εµ↓ + ωq
E − εµ↑ − ωq
µq
(12)
where nσµ = f (εµσ ) , ωq is the magnon energy, Nq = NB (ωq ) is the Bose function.
Using the expansion of the Dyson equation (10) we obtain for the spectral density
1
Aνσ (E) = − ImGσν (E) = δ(E − ενσ )
π
−δ ′ (E − ενσ )ReΣσν (E) −
1 ImΣσν (E)
π (E − ενσ )2
(13)
4
The second term in the right-hand side of Eq.(13) gives the shift of quasiparticle energies. The third term, which arises
from the branch cut of the self-energy, describes the incoherent (non-quasiparticle) contribution owing to scattering
by magnons. One can see that this does not vanish in the energy region, corresponding to the “alien” spin subband
with the opposite projection −σ.
Neglecting temporarily in Eq.(12) the magnon energy ωq in comparison with typical electron energies and using
the identities
Z
X | (νµ|q) |2
X | (νµ|q) |2
1
F (E ′ )
F (εµ ) = −
Im
(14)
dE ′
′
E − εµ
π
E−E
E ′ − εµ + i0
µq
µq
Z
X
1
F (E ′ )
= −
Im
hν| eiqr (E ′ − H0 + i0)−1 e−iqr |νi
dE ′
′
π
E−E
q
Z
′
1
F (E )
= −
Im hν| (E ′ − H0 + i0)−1 |νi
dE ′
π
E − E′
we derive
Σ↑ν (E) = 2I 2 S
Σ↓ν (E) = 2I 2 S
Z
Z
dE ′ f (E ′ ) hν ↑| δ E − E ′ − H0↓ |ν ↑i
dE [1 − f (E ′ )] hν ↓| δ E − E ′ − H0↑ |ν ↓i
(15)
(16)
Here we restrict ourselves only to the case of zero temperature T = 0 neglecting the magnon occupation numbers Nq .
Using the tight-binding model for the ideal-crystal Hamiltonian we find in the real-space representation
Z
1
Σ↑R,R′ (E) = 2I 2 S dE ′ f (E ′ ) − ImG↓R,R (E ′ ) δR,R′
(17)
π
Z
1
(18)
Σ↓R,R′ (E) = 2I 2 S dE ′ [1 − f (E ′ )] − ImG↑R,R (E ′ ) δR,R′
π
where R, R′ are lattice site indices, and therefore
Σσν (E) =
X
2
|ψνσ (R)| ΣσR,R (E).
(19)
R
Following the method developed by us earlier [12, 29] one can generalize the above results to the case of arbitrary
s−d exchange parameter. Simplifying the sequence of equations of motion (cf. Ref.29) we obtain the integral equation
X
σ
σ
(E) − σI
(E − εκ−σ )Rκλ
(E) Gσνκ (E)
(20)
(E − ενσ )Gσνλ (E) = δνλ + σIRνλ
κ
where
↑
Rνλ
(E) =
X
(µ ↓, λ ↑ | − q) (ν ↑, µ ↓ |q)
↓
Rνλ
(E) =
X
(µ ↑, λ ↓ |q) (ν ↓, µ ↑ |q)
µq
µq
n↓µ
,
E − εµ↓ + ωq
1 − n↑µ
E − εµ↑ − ωq
(21)
Note that the equation (20) is exact in the case of empty conduction band (nνσ = 0, one current carrier, ferromagnetic semiconductor situation), and for finite band filling this corresponds to a ladder approximation in the diagram
approach.
Similar to (14), we obtain after neglecting the magnon energies in (21) the equation for the Green’ function
X
σ
(E − εκ−σ )Rκλ
(E) Gσνκ (E) = hνσ| Rσ (E) (E − H0σ + i0)−1 Gσ (E) |νσi
(22)
κ
where we use the matrix notations. Then we have for the operator Green’ function
−1
1
−σ
σ
σ
σ
σ
G (E) = E − H0 + σI(H0 − H0 )
R (E)
1 + σIRσ (E)
(23)
5
If we consider spin dependence of electron spectrum in the simplest rigid-splitting approximation ενσ = εν − σI hS z i
and thus neglect spin-dependence of the eigenfunctions ψνσ (R) the expressions (15),(16) are drastically simplified.
Then the self-energy does not depend on ν and we have
2I 2 SRσ (E)
,
1 + σIRσ (E)
X n↓µ
X 1 − n↑µ
R↑ (E) =
, R↓ (E) =
E − εµ↓
E − εµ↑
µ
µ
Σσ (E) =
(24)
(25)
If H0σ is just the crystal Hamiltonian (ν = k, ενσ = tkσ , tkσ being the band energy), the expression (23) coincides
with that obtained in Ref. 12 for the Hubbard model after the replacement I → U .
The expression (23) can be also represented in the form
Gσ (E) = E − H0−σ − (H0σ − H0−σ )
1
1 + σIRσ (E)
−1
(26)
The equation (26) is convenient in the narrow-band case. In this limit where spin splitting is large in comparison with
the bandwidth of conduction electrons we have H0↑ − H0↑ = −2IS and we obtain for the “lower” spin subband with
σ = −signI
G (E) = E − H0−σ +
σ
2S
Rσ (E)
−1
(27)
2S
Rσ (E)
−1
(28)
For a periodic crystal Eq.(27) takes the form
Gσk (E) = E − tk−σ +
This expression yields exact result in the limit I → +∞,
−1
X 1 − f (tk )
2S
G↓k (E) = ǫ − tk +
, R(ǫ) =
R(ǫ)
ǫ − tk
(29)
k
with ǫ = E + IS, tk the bare electron spectrum. In the limit I → −∞ Eq.(28) gives correctly the spectrum of
spin-down quasiparticles,
G↓k (E) =
2S
−1
[ǫ − t∗k ]
2S + 1
(30)
with ǫ = E − I(S + 1), t∗k = [2S/(2S + 1)]tk . However, it does not describe the NQP states quite correctly, so that
more accurate expressions can be obtained by using the atomic representation [30],
G↑k
−1
X f (t∗ )
2S
2S
∗
k
(E) =
ǫ − tk + ∗
, R∗ (ǫ) =
2S + 1
R (ǫ)
ǫ − t∗k
(31)
k
On the other hand, the result (27), (28) yields a good interpolation description in the Hubbard model [9, 10, 12].
The Green’s functions (28), (29), (31) have no poles, at least for small current carrier concentration, and the whole
spectral weight of minority states is provided by the branch cut (non-quasiparticle states) [10, 12]. For surface states
this result was obtained in Ref.28 in a narrow-band Hubbard model. Now we see that this result can be derived in an
arbitrary inhomogeneous case. For a HMF with the gap in the minority spin subband NQP states occur above the
Fermi level, and for the gap in the majority spin subband below the Fermi level.
In the absence of spin dynamics (i.e., neglecting the magnon frequencies) the NQP density of states has a jump at
the Fermi level. However, the magnon frequencies can be restored in the final result, in analogy with the case of ideal
crystal, which leads to a smearing of the jump on the energy scale of a characteristic magnon energy ω. It should
be mentioned once more that we restrict ourselves to the case of the usual three-dimensional magnon spectrum and
do not consider the influence of surface states on the spin-wave subsystem. The expressions obtained enable us to
investigate the energy dependence of the spectral density.
6
IV.
THE NON-QUASIPARTICLE DENSITY OF STATES
An analysis of the electron-spin coupling yields different pictures for two possible signs of the s − d exchange
parameter I. For I > 0 the spin-down NQP scattering states form a “tail” of the upper spin-down band, which starts
from EF (Fig.1) since the Pauli principle prevents electron scattering into occupied states.
For I < 0 spin-up NQP states are present below the Fermi level as an isolated region (Fig.2): occupied states with
the total spin S − 1 are a superposition of the states |Si| ↓i and |S − 1i| ↑i. The entanglement of the states of electron
and spin subsystems which is necessary to form the NQP states is a purely quantum effect formally disappearing
at S → ∞. To understand better why the NQP states are formed only below EF in this case we can treat the
limit I = −∞. T hen the current carrier is really a many-body state of the occupied site as a whole with total spin
S − 1/2, which propagates in the ferromagnetic medium with spin S at any site. The fractions of the states |Si| ↓i
and |S − 1i| ↑i in the current carrier state are 1/(2S + 1) and 2S/(2S + 1), respectively, so that the first number
is just a spectral weight of occupied spin-up electron NQP states. At the same time, the density of empty states is
measured by the number of electrons with a given spin projection which one can add to the system. It is obvious
that one cannot put any spin-up electrons in the spin-up site at I = −∞. Therefore the density of NQP states should
vanish above EF .
It is worthwhile to note that in the most of known HMF the gap exists for minority-spin states [2]. This is similar
to the case I > 0, so that the NQP states should arise above the Fermi energy. For exceptional cases with the
majority-spin gap such as a double perovskite Sr2 FeMoO6 [31] one should expect formation of the NQP states below
the Fermi energy.
It has been proven in the previous section that the presence of space inhomogeneity (e.g., surface) does not change
qualitatively the spectral density picture, except smooth matrix elements. Therefore further in this section we will
consider, for simplicity, the case of clean infinite crystal; all the temperature and energy dependences of the spectral
density will be basically the same for the surface layer.
The second term in the right-hand side of Eq. (13) describes the renormalization of quasiparticle energies. The
third term, which arises from the branch cut of the self-energy Σνσ (E), describes the incoherent (non-quasiparticle)
contribution owing to scattering by magnons. One can see that this does not vanish in the energy region, corresponding
to the “alien” spin subband with the opposite projection −σ. Further on we perform for definiteness concrete
calculations in the case I > 0 (the case I < 0 differs, roughly speaking, by a particle-hole transformation). Summing
up Eq.(13) to find the total DOS Nσ (E) and neglecting the quasiparticle shift we obtain
X
2I 2 SNq
N↑ (E) =
1−
δ(E − tk↑ )
(tk+q↓ − tk↑ )2
kq
N↓ (E) = 2I 2 S
X
kq
1 + Nq − nk↑
δ(E − tk↑ − ωq )
(tk+q↓ − tk↑ − ωq )2
(32)
where we consider for simplicity only second-order perturbation expression. Basing on a general consideration in the
previous section one can prove that, actually, this expression holds for arbitrary I, at least, in the framework of 1/2S
expansion.
The T 3/2 -dependence of the magnon contribution to the residue of the Green’s function, i.e. of the effective electron
mass in the lower spin subband, and an increase with temperature of the incoherent tail from the upper spin subband
result in a strong temperature dependence of partial densities of states Nσ (E), the corrections being of opposite sign.
At the same time, the temperature shift of the band edge for the quasiparticle states is proportional to T 5/2 rather
than to magnetization [10, 29].
The exact solution in the atomic limit (for one conduction electron), which is valid not only in spin-wave region,
but for arbitrary temperatures, reads [32]
Gσ (E) =
S + 1 + σ hS z i
1
1
S − σ hS z i
+
.
2S + 1
E + IS
2S + 1 E − I (S + 1)
(33)
In this case the energy levels are not temperature dependent at all, whereas the residues are strongly temperature
dependent via the magnetization.
Now we consider the case T = 0 for a finite band filling. The picture of N (E) in HMF (or degenerate ferromagnetic
semiconductor) demonstrates strong energy dependence near the Fermi level (Figs. 1,2). If we neglect magnon
frequencies in the denominators of Eq.(32), the partial density of incoherent states should occur by a jump above or
below the Fermi energy EF for I > 0 and I < 0 respectively owing to the Fermi distribution functions. An account
of finite magnon frequencies ωq = Dq 2 (D is the spin wave stiffness constant) leads to smearing of these singularities,
N (EF ) being equal to zero. For |E − EF | ≪ ω we obtain
7
N−α (E)
1
=
Nα (E)
2S
E − EF 3/2
θ(α(E − EF )), α = sgnI
ω
(34)
(α = ± corresponds to the spin projections ↑, ↓). With increasing |E − EF |, N−α /Nα tends to a constant value which
is of order of I 2 within the perturbation theory.
In the strong coupling limit where |I| → ∞ we have from (32)
1
N−α (E)
=
θ(α(E − EF )), |E − EF | ≫ ω
Nα (E)
2S
(35)
In fact, this expression is valid only in the framework of the 1/2S-expansion, and in the narrow-band quantum case we
have to use more exact expressions (29),(31). In numerical calculations, we follow to Ref.30 and smear the resolvents,
Z
R(E) → R(E) = dωK(ω)R(E ± ω)
We use the semielliptic magnon DOS K(ω) which is proportional (with the corresponding shift) to the bare electron
DOS, the maximum magnon frequency being determined by the electron concentration c [30]. This approximation
provides the correct behavior near the Fermi level (cf. Ref.33), although gives an unphysical shift of the band bottom
by the maximum magnon frequency.
The results of calculations are shown in Figs.3, 4. One can see that in the model with I → −∞ (for S = 1/2 this
is equivalent to the Hubbard model with the replacement tk → tk /2, see Refs.10, 12) the “Kondo” peaks [33] modify
considerably the picture. Note that the function −(1/π)ImR∗ (E), which yields DOS in the lowest-order approximation
in the electron concentration, does not have such peaks.
To investigate details of the energy dependence of N (E) in the broad-band case we assume the simplest isotropic
approximation for the majority-spin electrons,
tk↑ − EF ≡ ξk =
k 2 − kF2
.
2m∗
(36)
Provided that we use the rigid splitting approximation tk↓ = tk↑ + ∆ (∆ = 2IS, I > 0), the half-metallic situation (or,
more precisely, the situation of degenerate ferromagnetic semiconductor) takes place for ∆ > EF . Then qualitatively
the equation (34) works to accuracy of a prefactor. It is worthwhile to note that, formally speaking, the NQP
contribution to DOS occurs also for an “usual” metal where ∆ < EF . In the case of small ∆ there is a crossover
energy (or temperature) scale
T ∗ = D (m∗ ∆/kF )2
(37)
which is the magnon energy at the boundary of Stoner continuum, T ∗ ≃ ω (∆/EF )2 ≪ ω. At |E − EF | ≪ ω the
equation (32) for the NQP contribution reads
"
#
p
1 1 + (E − EF ) /T ∗ p
p
δN↓ (E) ∝
(38)
ln − (E − EF ) /T ∗ θ(E − EF ).
2 1 − (E − EF ) /T ∗ ∗
At |E − EF | ≪ Tp
this gives the same results as above. However, at T ∗ ≪ |E − EF | ≪ ω this contribution is
proportional to − (E − EF ) /T ∗ and is negative (of course, the total DOS is always positive). This demonstrates
that one should be very careful when discussing the NQP states for the systems which are not half-metallic.
The model of rigid spin splitting used above is in fact not applicable for the real HMF where the gap has a
hybridization origin [1, 2]. The simplest model for HMF is as follows: a “normal” metallic spectrum for majority
electrons (36) and the hybridization gap for minority ones,
q
1
2
2
(39)
tk↓ − EF =
ξk + sgn (ξk ) ξk + ∆
2
Here we assume for simplicity that the Fermi energy lies exactly in the middle of the hybridization gap (otherwise one
needs to shift ξk → ξk + E0 − EF in the last equation, E0 being the middle of the gap). One can replace in Eq.(32)
ξk+q by vk q, vk = k/m∗ . First, we integrate over the angle between the vectors k and q. It is easy to calculate
*
h
2 +
i
1
8
2
3
2
3/2
X − (X + 1) + 1 + X
(40)
=
tk+q↓ − tk↑ − ωq
vF q∆ 3
8
where angular brackets stand for the average over the angles of the vector k, X = kF q/m∗ ∆. Here we do have the
crossover with the energy scale T ∗ which can be small for small enough hybridization gap. For example, in NiMnSb
the conduction band width is about 5 eV and the distance from the Fermi level to the nearest gap edge (i.e. indirect
energy gap which is proportional to ∆2 ) is smaller than 0.5 eV, so that (∆/EF )2 ≤ 0.1.
For the case 0 < E − EF ≪ ω one has
3/2
i
2 h 5/2
E − EF
y , y≪1
5/2
3/2
y
−
(1
+
y)
+
1
+
y
+
y
=
,
b(y)
=
(41)
N↓ (E) ∝ b
y, y ≫ 1
T∗
5
3/2
The function b(x) is shown in Fig.5. Thus the behavior N↓ (E) ∝ (E − EF )
takes place only for very small excitation
energies E −EF ≪ T ∗ , whereas in a broad interval T ∗ ≪ E −EF ≪ ω one has the linear dependence N↓ (E) ∝ E −EF .
V.
THE TEMPERATURE DEPENDENCE OF SPIN POLARIZATION
Simple qualitative considerations [34], as well as direct Green’s functions calculations [11, 35] for magnetic semiconductors, demonstrate that spin polarization of conduction electrons in the spin-wave region is proportional to
magnetization
P ≡
N↑ − N↓
= 2P0 hS z i
N↑ + N↓
(42)
A weak ground-state depolarization 1 − P0 occurs in the case where I > 0. The behavior P (T ) ≃ hS z i is qualitatively
confirmed by experimental data on field emission from ferromagnetic semiconductors [36] and transport properties of
half-metallic Heusler alloys [37].
An attempt was used [38] to generalize the result (42) on the HMF case (in fact, using qualitative arguments which
are valid only in the atomic limit, see Eq.(33)). However, we will demonstrate that the situation for HMF is more
complicated.
In this section we focus on the magnon contribution to DOS (32) and calculate the function
Λ=
X
kq
2I 2 SNq
δ(EF − tk↑ )
(tk+q↓ − tk↑ − ωq )2
(43)
Using the parabolic electron spectrum tk↑ = k 2 /2m∗ and averaging over the angles of the vector k we obtain
Λ=
2I 2 Sm2 X
Nq
ρ
∗ )2 − q 2
kF2
(q
q
(44)
where ρ = N↑ (EF , T = 0), we have used the condition q ≪ kF , q ∗ = m∗ ∆/kF = ∆/vF , where ∆ = 2 |I| S is the spin
splitting. In the ferromagnetic semiconductor we have, in agreement with the qualitative considerations presented
above:
3/2
T
S − hS z i
ρ∝
ρ
(45)
Λ=
2S
TC
Further on we consider the spectrum model (36), (39) where the gap has a hybridization origin. At T ≪ T ∗ we
reproduce the result (45) which is actually universal for this temperature region. At T ∗ ≪ T ≪ ω we derive
Λ=
X
kq
2I 2 SNq δ(ξk )
X Nq
16
T ∗1/2
T
∝ q∗
∝ 1/2 T ln ∗
3vF q∆
q
T
TC
q
(46)
This result distinguishes HMF like the Heusler alloys from ferromagnetic semiconductors and narrow-band saturated
ferromagnets. In the narrow-band case the spin polarization follows the magnetization up to the Curie temperature
TC .
For finite temperatures the density of NQP states at the Fermi energy is proportional to [11, 23, 34]
Z ∞
K(ω)
(47)
N (EF ) ∝
dω
sinh(ω/T
)
0
9
Generally, for temperatures which are comparable with the Curie temperature TC there are no essential difference
between half-metallic and “ordinary” ferromagnets since the gap is filled. The corresponding symmetry analysis is
performed in Ref. 23 for a model of conduction electrons interacting with “pseudospin” excitations in ferroelectric
semiconductors. The symmetrical (with respect to EF ) part of N (E) in the gap can be attributed to smearing of
electron states by electron-magnon scattering; the asymmetrical (“Kondo-like”) one is the density of NQP states owing
to the Fermi distribution function. Note that this filling of the gap is very important for possible applications of HMF
in spintronics: they really have some advantages only provided that T ≪ TC . Since a single-particle Stoner-like theory
leads to much less restrictive (but unfortunately completely wrong) inequality T ≪ ∆, the many-body treatment of
the spin-polarization problem (inclusion of collective spin-wave excitations) is required.
VI.
BIAS DEPENDENCE OF THE TUNNELING CONDUCTANCE
Now we consider an application of the results obtained above to the tunneling spectroscopy problem. The formulas
of Sect.4 for the energy dependence of NQP contributions are, strictly speaking, derived for the usual one-electron
density of states at EF , which is observed, say, in photoemission measurements. However, the factor of gsσ (E), which
is present in the expression for the tunneling current (4), does not influence the temperature dependence, and therefore
these results are valid for spin polarization from tunneling conductance at zero bias in STM.
The only difference in the NQP contributions to gsσ (E) and Nσ (E) is in that after summation over the magnon
wavevector q the integration is performed over not in the the whole Fermi surface, but its two points (see Eq.(5)). For
a spherical Fermi surface for majority electrons the results differ by the constant factor of the Fermi surface diameter.
However, the energy and temperature dependences should be the same in a more general case.
Consider the bias dependence of the tunneling current for zero temperature. One can see from Eq.(4) that
dI σ (V )
∝ gsσ (eV ) ∝ Nσ (eV )
dV
(48)
Again, the last proportionality can be strictly justified in the case of a spherical Fermi surface only, but is qualitatively
valid for arbitrary electron spectrum.
One should keep in mind that sometimes the surface of HMF is not half-metallic; in particular, this is the case of a
prototype HMF, NiMnSb [39]. In such a situation, the tunneling current for minority electrons is due to the surface
states only. However, the NQP states can be still visible in the tunneling current via the hybridization of the bulk
states with the surface one. The hybridization lead to the Fano antiresonance picture which is usually observed in
STM investigations of the Kondo effect at metallic surfaces (see, e.g., Refs. 40, 41, 42). In these cases the tunneling
conductance will be proportional to a mixture of Nσ (eV ) and Lσ (eV ), Lσ (E) being the real part of the on-site
Green’s function,
Z
Nσ (E ′ )
Lσ (E) = P dE ′
.
(49)
E − E′
(P stands for principal value, E is referred to the Fermi energy). Surprisingly, in this case the effect of NQP states
on the tunneling current can be even more pronounced in comparison with the ideal crystal. The reason is that the
analytical continuation of the jump in Nσ (E) is logarithm; both singularities are cut at the energy ω; nevertheless,
the energy dependence of Lσ (E) can be pronounced, see Fig.6. This is similar to the effect of enhancement of the
NQP contribution to the x-ray absorption and emission spectra, which was predicted in Ref. 15.
Now we discuss in more detail the energy dependence for |E| ≪ ω. The analytical continuation of the E 3/2 θ(E)contribution to Nσ (E) yields the contribution (−E)3/2 θ(−E) in Lσ (E) which is non-zero on the other side with
respect to EF (a situation that is formally similar to the electronic topological transition, see Ref.46). The one-sided
linear dependence in Nσ (E) according to Eq.(41) corresponds to E ln |E| in Lσ (E).
STM measurements of electron DOS give also an opportunity to probe bosonic excitations interacting with the
conduction electrons. Due to electron-phonon coupling, the derivative dNσ (E) /dE and thus d2 I σ (V ) /dV 2 at eV = E
have peaks at the energies E = ±ωi corresponding to the peaks in the phonon DOS. According to our results (see,
e.g., Eq.(32), the same effect should be observable for the case of electron-magnon interaction. However, in the latter
case these peaks are essentially asymmetric with respect to the Fermi energy (zero bias) due to asymmetry of the
non-quasiparticle contributions. This asymmetry can be used to distinguish phonon and magnon peaks.
10
VII.
CONCLUSIONS
In the present paper we have demonstrated that non-quasiparticle states in half-metallic ferromangnets exist not
only for an ideal crystal, but also in the presence of an arbitrary external potential. In particular, they occur at the
surface of the half-metallic ferromagnets. These states can be probed by the STM both directly and via their effect on
the surface states (the Fano antiresonance case). Therefore, they can be observable even for the situation of surface
“dead layers” where the surface is not half-metallic. The expressions obtained can be used for realistic electronic
structure calculations of NQP contributions to the electron energy spectrum of the surfaces of HMF.
Temperature dependence of the spin polarization at the Fermi energy which can be also probed by the STM follows
the temperature dependence of magnetization an very low temperatures. For the HMF with a hybridization gap,
there is a crossover energy (temperature) T ∗ ≪ TC where the character of the temperature dependence is changed.
The energy dependence of the NQP contributions (and consequently the bias dependence of the tunneling current)
is strongly influenced by the band structure too. In particular, for HMF with a hybridization gap this demonstrates
a linear rather than E 3/2 behavior in a wide interval. In the narrow-band case a Kondo-like peak (Fig.4) near the
Fermi level should be observed in tunneling experiments.
Due to asymmetry of NQP states with respect to the Fermi energy, the magnon peaks in d2 I σ (V ) /dV 2 are also
asymmetric with respect to the zero bias, in contrast with the phonon ones. This gives an opportunity to distinguish
between phonon and magnon peaks in the inelastic spectroscopy by STM.
In principle, the NQP effects discussed should exist also in usual metallic ferromagnets. However, only in HMF
they can be picked up in a pure form.
The research described was supported in part by Grant No. 747.2003.2 (Support of Scientific Schools) from the
Russian Basic Research Foundation and by the Netherlands Organization for Scientific Research (Grant NWO
047.016.005).
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12
N E -0.1
0.1
N
0.2
0.3
E
FIG. 1: Density of states in the s − d exchange model of a half-metallic ferromagnet with S = 1/2, I = 0.3 for the semielliptic
bare band with the width of W = 2. The Fermi energy calculated from the band bottom is 0.15 (the energy is referred to EF ).
The magnon band is also assumed semielliptic with the width of ωmax = 0.02. The non-quasiparticle tail of the spin-down
subband (lower half of the figure) occurs above the Fermi level. The corresponding picture for the empty conduction band is
shown by dashed line; the short-dashed line corresponds to the mean-field approximation.
N E -0.1
0.1
0.2
N E FIG. 2: Density of states in a half-metallic ferromagnet with I = −0.3 < 0, other parameters being the same as in Fig.1. The
spin-down subband (lower half of the figure) nearly coincides with the bare band shifted by IS. Non-quasiparticle states in the
spin-up subbands (upper half of the figure) occur below the Fermi level; the short-dashed line corresponds to the mean-field
approximation.
N E -0.1
0.1
0.3
0.5
N E FIG. 3: Density of states in a half-metallic ferromagnet in the s-d model with I → +∞, S = 1/2. The Fermi energy calculated
from the bare band bottom is 0.1 (concentration of conduction electrons is c = 0.019). The dashed line shows the function
−(1/π)ImR(E).
13
N E -0.4
-0.3
-0.2
-0.1
N
E
FIG. 4: Density of states in a half-metallic ferromagnet in the s-d model with I → −∞, S = 1/2.The Fermi energy calculated
from the bare band bottom is 0.2 (c = 0.034). The dashed line shows the function −(1/π)ImR∗ (E).
b x 2
1.5
1
0.5
x
0.5
1
1.5
2
2.5
3
FIG. 5: Plot of the function b(x).
-0.02
0.02
0.04 0.06 0.08
0.1
N E L E FIG. 6: Plot of the imaginary (upper line) and real (lower line) parts of the Green’s function near the Fermi level in a
half-metallic ferromagnet with the same parameters as in Fig.1.